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Claim: Suppose that $X,Y$ are independent random variables, then $X$ and $1/Y$ are also independent random variables.

There are hints pointing to that this claim is true

I have no idea how to prove this. But let's try.

Suppose $X$, $Y$ are independent, then $P[X = x, Y = y] = P[X = x]P[Y= y]$, we wish to show that $P[X = x, 1/Y = y] = P[X = x]P[1/Y= y]$. Not obvious how to proceed from this point.

Hmm...try another definition.

Suppose $X$, $Y$ are independent, then $P[X| Y = y] = P[X = x] = P[X|Y = 1/y] = P[X| 1/Y = y]$, therefore $X, 1/Y$ are independent.

Would the second attempt hold?

Is there any generalization to the above claim?

Norman
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  • https://math.stackexchange.com/questions/474202/criterion-for-independency-of-random-variables?rq=1 –  Oct 10 '17 at 20:10

1 Answers1

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Yes the second attempt looks fine.

You can proceed with the first attempt: Define $U=1/Y$ as a new variable to avoid confusion.

$$P(X=x,Y=y)=P(X=x)P(Y=y)\\\iff P(X=x,1/Y=1/y)=P(X=x)P(1/Y=1/y)\\\iff P(X=x,U=u)=P(X=x)P(U=u)$$ where $u=1/y$. So one can conclude that $X$ and $U$ are independent, as required.

John Doe
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